OFFSET
1,1
COMMENTS
Horizontally or vertically adjacent entries can differ by at most 1. Diagonally adjacent entries thus differ by at most 2.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
Michael Han, Sycamore Herlihy, Kirsti Kuenzel, Daniel Martin, and Rachel Schmidt, The number of independent sets in bipartite graphs and benzenoids, arXiv:2311.15334 [math.CO], 2023. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (7,-4).
FORMULA
a(n)=b(n)+c(n), where b(1)=2, c(1)=1, b(n+1)=4*b(n)+4*c(n), c(n+1)=2*b(n)+3*c(n).
G.f.: x*(3 - 2*x) / (1 - 7*x + 4*x^2). - N. J. A. Sloane, Apr 06 2008
a(n+2) = 7*a(n+1) - 4*a(n) for n >= 2. - Robert Israel, Sep 02 2014
a(n) = (2^(-2-n)*((7-sqrt(33))^n*(-5+sqrt(33)) + (5+sqrt(33))*(7+sqrt(33))^n)) / sqrt(33). - Colin Barker, Jan 31 2018
EXAMPLE
a(1) = 3:
|1|1|1|
|0|1|2|
a(2) = 19:
|10|11|12| |10|11|12| |10|11|12|
|0*|0*|01| |1*|1*|1*| |21|2*|2*|
(3) (2)(1) (2) (3)(2) (1) (2)(3), total 19.
MAPLE
a:= LREtools[REtoproc](a(n+3)=7*a(n+2)-4*a(n+1), a(n), {a(0)=0, a(1)=3, a(2)=19}):
seq(a(n), n=1..100); # Robert Israel, Sep 02 2014
MATHEMATICA
LinearRecurrence[{7, -4}, {3, 19}, 22] (* Jean-François Alcover, Apr 30 2019 *)
PROG
(PARI) Vec(x*(3 - 2*x) / (1 - 7*x + 4*x^2) + O(x^30)) \\ Colin Barker, Jan 31 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wayne VanWeerthuizen, Apr 05 2008
STATUS
approved